Cosets and Lagrange's Theorem

Question

This question is under the topic of Cosets and lagrange theorem. Now Is it true that if $G$ is a group that contains a subgroup $H_1$ of order $n$ and a subgroup $H_2$ of order $k$, then $G$ must contain a subgroup of order $nk$? Really need an example for this proof

score 2 · Answer 1 · answered Sep 12 '13 at 03:34

2

No; the easiest counterexample is to take a group $G$ of order $n > 1$, letting $H_1 = H_2 = G$.

answered Sep 12 '13 at 03:34

score 0 · Answer 2 · answered Sep 13 '13 at 09:10

0

Your result would hold if $G$ was abelian and the orders of $H_{1}$ and $H_{2}$ were coprime, but that is a fairly restrictive condition!

answered Sep 13 '13 at 09:10

David Ward

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Cosets and Lagrange's Theorem

2 Answers2